Below you will find **course descriptions** and **syllabi** for all classes offered by the Applied Mathematics department. Not every class is offered each semester, so we have a tentative schedule of when classes are expected to be offered. We link to that schedule and other information useful for planning your future, as well as information for current courses, just below.

For courses offered by the Applied Mathematics Department:

- Tentative Future Schedule of Course Offerings (through Spring 2019)
- Textbooks (Spring 2013)
- Teaching Assistant Office Hours in E1 129 (Spring 2015)
- Web pages (Spring 2013), but most use Blackboard instead
- Undergraduate Handbook for Applied Mathematics (pdf) (updated Fall 2013)

For information about courses across all of IIT, including course sections available in the upcoming semester (when ready):

- Course Schedule/Search by semester, department, etc.
- Undergraduate Bulletin (2014 - 2016)
- Graduate Bulletin (2014 - 2016)

## Undergraduate

### MATH 100 Introduction to the Profession

Introduces the students to the scope of mathematics as a profession, develops a sense of mathematical curiosity and problem solving skills, identifies and reinforces the student's career choices, and provides a mechanism for regular academic advising. Provides integration with other first-year courses. Introduces applications of mathematics to areas such as engineering, physics, computer science, and finance. Emphasis is placed on the development of teamwork skills. (3-0-3)

**» Sample Syllabus**

### MATH 119 Geometry for Architects

Basic analytic geometry in two and three dimensions; trigonometry. Equations of lines, circles and conic sections; resolution of triangles; polar coordinates. Equations of planes, lines, and quadratic surfaces. Applications. (3-0-3) (C)

This course does not count for graduation in any engineering, mathematics, natural science or computer science degree program.

**» Sample Syllabus**

### MATH 122 Introduction to Calculus

Basic concepts of calculus of a single variable; limits, derivatives, integrals. Applications. (3-0-3)

Credit may not be granted for both MATH 122 and MATH 123. This course does not count for graduation in any engineering, mathematics, natural science or computer science degree program.

**» Sample Syllabus**

### MATH 130 Thinking Mathematically

This course allows students to discover, explore and apply modern mathematical ideas. Emphasis is placed on using sound reasoning skills, visualization of mathematical concepts and on communicating mathematical ideas effectively. Classroom discussion and group work on challenging problems are central to the course. Topics from probability, statistics, logic, number theory, graph theory, combinatorics, chaos theory, the concept of infinity, and geometry may be included. (3-0-3) (C)

**» Sample Syllabus**

### MATH 148 Calculus/Precalculus I

Review of algebra and analytic geometry. Functions, limits, derivatives. Trigonometry, trigonometric functions and their derivatives. Chain rule, implicit and inverse functions, and inverse trigonometric functions. (4-0-4)

This course does not count for graduation in any engineering, mathematics, natural science or computer science degree program.

**» Sample Syllabus**

### MATH 149 Calculus/Precalculus II

Applications of derivatives; related rates, maxima and minima, monotonicity, concavity, graphing, optimization. Antiderivatives, first-order differential equations. Definite integrals and applications. Implicit and inverse functions, and inverse trigonometric functions. (4-1-5) (C)

**» Sample Syllabus**

### MATH 151 Calculus I

Analytic geometry. Functions and their graphs. Limits and continuity. Derivatives of algebraic, trigonometric and inverse trigonometric functions. Applications of the derivative. Introduction to integrals and their applications. (4-1-5) (C)

Prerequisites: Must pass departmental pre-calculus placement exam.

**» Sample Syllabus**

### MATH 152 Calculus II

Transcendental functions and their calculus. Integration techniques. Applications of the integral. Indeterminate forms and improper integrals. Polar coordinates. Numerical series and power series expansions. (4-1-5) (C)

Prerequisites: Grade of "C" or better in MATH 151 or MATH 149 or Advanced Placement.

**» Sample Syllabus**

### MATH 230 Introduction to Discrete Mathematics

Sets, statements, and elementary symbolic logic; relations and digraphs; functions and sequences; mathematical induction; basic counting techniques and recurrence. (3-0-3) (C)

Credit will not be granted for both CS 330 and MATH 230.

**» Sample Syllabus**

### MATH 251 Multivariate and Vector Calculus

Analytic geometry in three-dimensional space. Partial derivatives. Multiple integrals. Vector analysis. Applications. (4-0-4)

Prerequisite: MATH 152.

**» Sample Syllabus**

### MATH 252 Introduction to Differential Equations

Linear differential equations of order one. Linear differential equations of higher order. Series solutions of linear DE. Laplace transforms and their use in solving linear DE. Introduction to matrices. Systems of linear differential equations. (4-0-4)

Prerequisite: MATH 152.

**» Sample Syllabus**

### MATH 300 Perspectives in Analysis

The course is focused on selected topics related to fundamental concepts and methods of classic analysis and their applications with emphasis on various problem-solving strategies, visualization, mathematical modeling, and interrelation of different areas of mathematics. To be cross-listed with MSED 521. (3-0-3)

Prerequisites: MATH 251 and MATH 252 or consent of the instructor.

**» Sample Syllabus**

### MATH 332 Elementary Linear Algebra

Systems of linear equations; matrix algebra, inverses, determinants, eigenvalues and eigenvectors, diagonalization; vector spaces, basis, dimension, rank and nullity; inner product spaces, orthonormal bases; quadratic forms. (3-0-3)

Prequisite: MATH 251.

**» Sample Syllabus**

### MATH 333 Matrix Algebra and Complex Variables

Vectors and matrices; matrix operations, transpose, rank, inverse; determinants; solution of linear systems; eigenvalues and eigenvectors. The complex plane; analytic functions; contour integrals; Laurent series expansions; singularities and residues. (3-0-3)

Not applicable for Math majors.

Prerequisite: MATH 251.

**» Sample Syllabus**

### MATH 350 Introduction to Computational Mathematics

Study and design of mathematical models for the numerical solution of scientific problems. This includes numerical methods for the solution on linear and nonlinear systems, basic data fitting problems, and ordinary differential equations. Robustness, accuracy, and speed of convergence of algorithms will be investigated including the basics of computer arithmetic and round-off errors. Same as MMAE 350. (3-0-3)

Prerequisites: MATH 251, MATH 252, and CS 104. CS105 or CS 115, or consent of instructor.

**» Sample Syllabus**

### MATH 374 Probability and Statistics for Electrical and Computer Engineers

This course focuses on the introductory treatment of probability theory including: axioms of probability, discrete an continuous random variables, random vectors, marginal, joint, conditional and cumulative probability distributions, moment generating functions, expectations, and correlations. Also covered are sums of random variables, central limit theorem, sample means, and parameter estimation. Furthermore, random processes and random signals are covered. Examples and applications are drawn from problems of importance to electrical and computer engineers. (3-0-3)

Prerequisites: MATH 251.

**» Sample Syllabus**

### MATH 380 Introduction to Mathematical Modeling

This course provides an introduction to problem-driven (as opposed to method-driven) applications of mathematics with a focus on design and analysis of models using tools from all parts of mathematics. (3-0-3)

Prerequisites: CS 104, MATH 251, MATH 252 (concurrent), MATH 332.

**» Sample Syllabus**

### MATH 400 Real Analysis

Real numbers, continuous functions; differentiation and Riemann integration. Functions defined by series. (3-0-3)

Prerequisite: MATH 251 or consent of instructor.

**» Sample Syllabus**

### MATH 402 Complex Analysis

Analytic functions, conformal mapping, contour integration, series expansions, singularities and residues, and applications. Intended as a first course in the subject for students in the physical sciences and engineering. (3-0-3)

Prerequisite: MATH 251.

**» Sample Syllabus**

### MATH 405 Introduction to Iteration and Chaos

Functional iteration and orbits, periodic points and Sharkovsky's cycle theorem, chaos and dynamical systems of dimensions one and two. Julia sets and fractals, physical implications. (3-0-3) (C)

Prerequisites: MATH 251, MATH 252 and one of the following: MATH 332, MATH 333, or consent of the instructor.

**» Sample Syllabus**

### MATH 410 Number Theory

Divisibility, congruences, distribution of prime numbers, functions of number theory, diophantine equations, applications to encryption methods. (3-0-3)

Prerequisite: MATH 230 or consent of instructor.

**» Sample Syllabus**

### MATH 420 Geometry

The course is focused on selected topics related to fundamental concepts and methods of Euclidean geometry in two and three dimensions and their applications with emphasis on various problem-solving strategies, geometric proof, visualization, and interrelation of different areas of mathematics. To be cross-listed with MSED 520. (3-0-3)

Prerequisite: Consent of the instructor.

**» Sample Syllabus**

### MATH 425 Statistical Methods

Concepts and methods of gathering, describing and analyzing data including basic statistical reasoning, basic probability, sampling, hypothesis testing, confidence intervals, correlation, regression, forecasting, and nonparametric statistics. No knowledge of calculus is assumed. This course is useful for students in education or the social sciences. (3-0-3)

Credit given only for one of MATH 425, MATH 476 or MATH 525. This course does not count for graduation in any mathematics programs.

**» Sample Syllabus**

### MATH 426 Statistical Tools for Engineers

Descriptive statistics and graphs, probability distributions, random sampling, independence, significance tests, design of experiments, regression, time-series analysis, statistical process control, introduction to multivariate analysis. Same as CHE 426. (3-0-3)

Prerequisite: Junior standing.

### MATH 430 Applied Algebra

Introduction to groups, homomorphisms, group actions, rings, field theory. Applications, including constructions with ruler and compass, solvability by radicals, error correcting codes. (3-0-3)

Prerequisite: MATH 230. Concurrent Prerequisite: MATH 332.

**» Sample Syllabus**

### MATH 431 Computational Algebraic Geometry

Systems of polynomial equations and ideals in polynomial rings; solution sets of equations and algebraic varieties in affine n-space; effective manipulation of ideals and varieties, algorithms for basic algebraic computations; Groebner bases; applications. (3-0-3)

Credit may not be granted for both MATH 431 and MATH 530.

Prerequisite: MATH 332

**» Sample Syllabus**

### MATH 435 Linear Optimization

Introduction to both theoretical and algorithmic aspects of linear optimization: geometry of linear programs, simplex method, anticycling, duality theory and dual simplex method, sensitivity analysis, large scale optimization via Dantzig-Wolfe decomposition and Benders decomposition, interior point methods, network flow problems, integer programming. (3-0-3)

Credit may not be granted for both MATH 435 and MATH 535.

Prerequisite: MATH 230 or MATH 332.

**» Sample Syllabus**

### MATH 453 Combinatorics

Permutations and combinations; pigeonhole principle; inclusion-exclusion principle; recurrence relations and generating functions; enumeration under group action. (3-0-3)

Prerequisite: MATH 230 or consent of instructor.

**» Sample Syllabus**

### MATH 454 Graph Theory and Applications

Directed and undirected graphs; paths, cycles, trees, Eulerian cycles, matchings and coverings, connectivity, Menger's Theorem, network flow, coloring, planarity, with applications to the sciences (computer, life, physical, social) and engineering. (3-0-3) (C)

Prerequisite: MATH 230 and (MATH 251 or MATH 252)

**» Sample Syllabus**

### MATH 461 Fourier Series and Boundary-Value Problems

Fourier series and integrals. The Laplace, heat, and wave equations: Solutions by separation of variables. D'Alembert's solution of the wave equation. Boundary-value problems. (3-0-3)

Prerequisites: MATH 251, MATH 252.

**» Sample Syllabus**

### MATH 474 Probability and Statistics

Elementary probability theory including discrete and continuous distributions, sampling, estimation, confidence intervals, hypothesis testing, and linear regression. (3-0-3)

Not applicable for applied math majors. Credit not granted for both MATH 474 and MATH 475.

Prerequisite: MATH 251.

**» Sample Syllabus**

### MATH 475 Probability

Elementary probability theory; combinatorics; random variables; discrete and continuous distributions; joint distributions and moments; transformations and convolution; basic theorems; simulation. (3-0-3)

Credit not granted for both MATH 474 and MATH 475.

Prerequisite: MATH 251.

**» Sample Syllabus**

### MATH 476 Statistics

Estimation theory; hypothesis tests; confidence intervals; goodness-of-fit tests; correlation and linear regression; analysis of variance; nonparametric methods. (3-0-3)

Credit given only for one of MATH 425, MATH 476 or MATH 525.

Prerequisite: MATH 475.

**» Sample Syllabus**

### MATH 477 Numerical Linear Algebra

Fundamentals of matrix theory; least squares problems; computer arithmetic; conditioning and stability; direct and iterative methods for linear systems; eigenvalue problems. (3-0-3)

Prerequisite: MATH 350 or MMAE 350, or consent of the instructor.

**» Sample Syllabus**

### MATH 478 Numerical Methods for Differential Equations

Polynomial interpolation; numerical integration; numerical solution of initial value problems for ordinary differential equations by single and multi-step methods, Runge-Kutta, Predictor-Corrector; numerical solution of boundary value problems for ordinary differential equations by shooting method, finite differences and spectral methods. (3-0-3)

Prerequisite: MATH 350 or MMAE 350, or consent of the instructor.

**» Sample Syllabus**

### MATH 481 Introduction to Stochastic Processes

This is an introductory course in stochastic processes. Its purpose is to introduce students to a range of stochastic processes which are used as modeling tools in diverse fields of applications, especially in the business applications. The course introduces the most fundamental ideas in the area of modeling and analysis of real World phenomena in terms of stochastic processes. The course covers different classes of Markov processes: discrete and continuous-time Markov chains, Brownian motion and diffusion processes. It also presents some aspects of stochastic calculus with emphasis on the application to financial modeling and financial engineering. (3-0-3)

Credit may not be granted for MATH 481 and MATH 542.

**» Sample Syllabus**

### MATH 483 Design and Analysis of Experiments

Review of elementary probability and statistics; analysis of variance for design of experiments; estimation parameters; confidence intervals for various linear combinations of the parameters; selection of sample sizes; various plots of residuals; block designs; Latin Squares; one, two and 2^{k} factorial designs; nested and cross factor designs; regression; nonparametric techniques. (3-0-3)

Prerequisites: MATH 476 or MATH 474.

**» Sample Syllabus**

### MATH 484 Regression and Forecasting

Simple linear regression; multiple linear regression; least squares estimates of parameters; hypothesis testing and confidence intervals in linear regression models; testing of models, data analysis and appropriateness of models; linear time series models; moving average, autoregressive and/or ARIMA models; estimation, data analysis and forecasting with time series models; forecasting errors and confidence intervals.

**» Sample Syllabus**

### MATH 485 Introduction to Mathematical Finance

This is an introductory course in mathematical finance. Technical difficulty of the subject is kept at a minimum by considering a discrete time framework. Nevertheless, the major ideas and concepts underlying modern mathematical finance and financial engineering will be explained and illustrated. (3-0-3)

Credit may not be granted for MATH 485 and MATH 548.

Prerequisite: MATH 475 or equivalent.

**» Sample Syllabus**

### MATH 486 Mathematical Modeling I

This course provides a systematic approach to modeling and analysis of physical processes. For specific applications, relevant differential equations are derived from basic principles, for example from conservation laws and constitutive equations. Dimensional analysis and scaling are introduced to prepare a model for analysis. Analytic solution techniques, such as integral transforms and similarity variable techniques, or approximate methods, such as asymptotic and perturbation methods, are presented and applied to the models. A broad range of applications from areas such as physics, engineering, biology, and chemistry are studied. (3-0-3) (C)

Credit may not be granted for both MATH 486 and MATH 522.

Prerequisites: MATH 461

**» Sample Syllabus**

### MATH 487 Mathematical Modeling II

The formulation of mathematical models, solution of mathematical equations, interpretation of results. Selected topics from queuing theory and financial derivatives. (3-0-3) (C)

### MATH 488 Ordinary Differential Equations and Dynamical Systems

Boundary-value problems and Sturm-Liouville theory; linear system theory via eigenvalues and eigenvectors; Floquet theory; nonlinear systems: critical points, linearization, stability concepts, index theory, phase portrait analysis, limit cycles, and stable and unstable manifolds; bifurcation; and chaotic dynamics. (3-0-3)

**» Sample Syllabus**

### MATH 489 Partial Differential Equations

First-order equations, characteristics. Classification of second-order equations. Laplace's equation; potential theory. Green's function, maximum principles. The wave equation: characteristics, general solution. The heat equation: use of integral transforms. (3-0-3)

Prerequisite: MATH 461.

**» Sample Syllabus**

### MATH 491 Reading and Research

(Credit: Variable) (C)

### MATH 497 Special Problems

(Credit Variable) (C)

**» Sample Syllabus**

## Graduate

### MATH 500 Applied Analysis I

Measure Theory and Lebesgue Integration; Metric Spaces and Contraction Mapping Theorem, Normed Spaces; Banach Spaces; Hilbert Spaces. (3-0-3)

Prerequisite: MATH 400 or consent of the instructor.

**» Sample Syllabus**

### MATH 501 Applied Analysis II

Bounded Linear Operators on a Hilbert Space; Spectrum of Bounded Linear Operators; Fourier Series; Linear Differential Operators and Green's Functions; Distributions and the Fourier Transform; Differential Calculus and Variational Methods. (3-0-3)

Prerequisite: MATH 500 or consent of the instructor.

**» Sample Syllabus**

### MATH 512 Partial Differential Equations

Basic model equations describing wave propagation, diffusion and potential functions; characteristics, Fourier transform, Green function, and eigenfunction expansions; elementary theory of partial differential equations; Sobolev spaces; linear elliptic equations; energy methods; semigroup methods; applications to partial differential equations from engineering and science. Offered every two years. (3-0-3)

Prerequisites: MATH 461 or MATH 489 or consent of the instructor.

**» Sample Syllabus**

### MATH 515 Ordinary Differential Equations and Dynamical Systems

Basic theory of systems of ordinary differential equations; equilibrium solutions, linerization and stability; phase portraits analysis; stable unstable and center manifolds; periodic orbits, homoclinic and heteroclinic orbits; bifurcations and chaos; nonautonomous dynamics; and numerical simulation of nonlinear dynamics. (3-0-3)

Prerequisite: MATH 252 or consent of the instructor.

**» Sample Syllabus**

### MATH 519 Complex Analysis

Analytic functions, contour integration, singularities, series, conformal mapping, analytic continuation, multivalued functions. (3-0-3)

Prerequisite: MATH 402 or instructor's consent.

**» Sample Syllabus**

### MATH 522 Mathematical Modeling

This course provides a systematic approach to modeling and analysis of physical processes. For specific applications, relevant differential equations are derived from basic principles, for example from conservation laws and constitutive equations. Dimensional analysis and scaling are introduced to prepare a model for analysis. Analytic solution techniques, such as integral transforms and similarity variable techniques, or approximate methods, such as asymptotic and perturbation methods, are presented and applied to the models. A broad range of applications from areas such as physics, engineering, biology, and chemistry are studied. (3-0-3) (C)

Credit may not be granted for both MATH 486 and MATH 522.

Prerequisites: MATH 461

**» Sample Syllabus**

### MATH 523 Case Studies and Project Design in Applied Mathematics

This course is a Capstone course for the Master of Applied Mathematics program. Students will work in small groups on case studies of real world problems. A survey of applied mathematics of particular importance to real world is included, if needed. Students will gain experience in team projects in applied mathematics, including report generation and oral presentations. (6-0-6)

Prerequisites: COM523 or SCI 522, CHEM511, MATH522

### MATH 525 Statistical Models and Methods

Concepts and methods of gathering, describing and analyzing data including statistical reasoning, basic probability, sampling, hypothesis testing, confidence intervals, correlation, regression, forecasting, and nonparametric statistics. No knowledge of calculus is assumed. This course is useful for graduate students in education or the social sciences. (3-0-3)

Credit given only for one of MATH 425, MATH 476 or MATH 525. This course does not count for graduation in any mathematics program.

**» Sample Syllabus**

### MATH 530 Applied and Computational Algebra

Basics of computation with systems of polynomial equations, ideals in polynomial rings; solving systems of equations by Groebner bases; introduction to elimination theory; algebraic varieties in affine n-space; Zariski topology; dimension, degree, their computation and theoretical consequences. (3-0-3)

Prerequisite: MATH 332 or MATH 532.

**» Sample Syllabus**

### MATH 532 Linear Algebra

Matrix algebra, vector spaces, norms, inner products and orthogonality, determinants, linear transformations, eigenvalues and eigenvectors, Cayley-Hamilton theorem, matrix factorizations (LU,QR, SVD). (3-0-3)

Prerequisite: Undergraduate linear algebra (MATH 332) or instructor’s consent.

**» Sample Syllabus**

### MATH 535 Optimization I

Introduction to both theoretical and algorithmic aspects of linear optimization: geometry of linear programs, simplex method, anticycling, duality theory and dual simplex method, sensitivity analysis, large scale optimization via Dantzig-Wolfe decomposition and Benders decompostion, interior point methods, network flow problems, integer programming. (3-0-3)

Credit may not be granted for both MATH 435 and MATH 535.

Prerequisite: Undergraduate linear algebra (MATH 332) or instructor’s consent.

**» Sample Syllabus**

### MATH 540 Probability

Random events and variables, probability distributions, sequences of random variables and limit theorems. (3-0-3)

Prerequisites: MATH 400, MATH 475 or consent of the instructor.

**» Sample Syllabus**

### MATH 542 Stochastic Processes

This is an introductory course in stochastic processes. Its purpose is to introduce students into a range of stochastic processes, which are used as modeling tools in diverse field of applications, especially in the business applications. The course introduces the most fundamental ideas in the area of modeling and analysis of real World phenomena in terms of stochastic processes. The course covers different classes of Markov processes: discrete and continuous-time Markov chains, Brownian motion and diffusion processes. It also presents some aspects of stochastic calculus with emphasis on the application to financial modeling and financial engineering. (3-0-3)

Credit may not be granted for MATH 481 and MATH 542.

Prerequisites: MATH 332 or 333 or equivalent; MATH 475.

**» Sample Syllabus**

### MATH 543 Stochastic Analysis

This course will introduce the student to modern finite dimensional stochastic analysis and its applications. The topics will include: a) an overview of modern theory of stochastic processes, with focus on semimartingales and their characteristics, b) stochastic calculus for semimartingales, including Ito formula and stochastic integration with respect to semimartingales, c) stochastic differential equations (SDE's) driven by semimartingales, with focus on stochastic SDE's driven by Levy processes, d) absolutely continuous changes of measures for semimartingales, e) some selected applications. (3-0-3) Prerequisite: MATH 540 or consent of an instructor.

**» Sample Syllabus**

### MATH 544 Stochastic Dynamics

This course is about modeling, analysis, simulation and prediction of dynamical behavior of complex systems under random influences. The mathematical models for such systems are in the form of stochastic differential equations. It is especially appropriate for graduate students who would like to use stochastic methods in their research, or to learn these methods for long term career development. Topics include white noise and colored noise, stochastic differential equations, random dynamical systems, numerical simulation, and applications to scientific, engineering and other areas. (3-0-3)

Prerequisites: MATH 474 or MATH 475 or equivalent.

**» Sample Syllabus**

### MATH 545 Stochastic Partial Differential Equations

This course introduces various methods for understanding solutions and dynamical behaviors of stochastic partial differential equations arising from mathematical modeling in science and engineering and other areas. It is designed for graduate students who would like stochastic methods in their research or to learn such methods for long term career development. Topics include: Random variables, Brownian motion and stochastic calculus in Hilbert spaces; Stochastic heat equation; Stochastic wave equation; Analytical and approximation techniques; Stochastic numerical simulations via Matlab; Dynamical impact of noises; Stochastic flows and cocycles; Invariant measures, Lyapunov exponents and ergodicity; and applications to engineering and science and other areas. Offered every two years. (3-0-3)

Prerequisites: MATH 543 or 544 or consent of the instructor.

**» Sample Syllabus**

### MATH 546 Introduction to Time Series

Properties of stationary, random processes; standard discrete parameter models, autoregressive, moving average, harmonic; standard continuous parameter models. Spectral analysis of stationary processes, relationship between the spectral density function and the autocorrelation function; spectral representation of some stationary processes; linear transformations and filters. Introduction to estimation in the time and frequency domains. (3-0-3)

Prerequisite: MATH 475 or ECE 511.

### MATH 548 Mathematical Finance I

This is an introductory course in mathematical finance. Technical difficulty of the subject is kept at a minimum by considering a discrete time framework. Nevertheless, the major ideas and concepts underlying modern mathematical finance and financial engineering are explained and illustrated. (3-0-3)

Credit may not be granted for MATH 485 and MATH 548.

Prerequisite: MATH 475 or equivalent.

**» Sample Syllabus**

### MATH 550 Topology

Topological spaces, continuous mappings and homoeomorphisms, metric spaces and metrizability, connectedness and compactness, homotopy theory. (3-0-3) Prerequisite: MATH 556.

**» Sample Syllabus**

### MATH 553 Discrete Applied Mathematics I

A graduate-level introduction to modern graph theory through existential and algorithmic problems, and the corresponding structural and extremal results from matchings, connectivity, planarity, coloring, Turan-type problems, and Ramsey theory. Proof techniques based on induction, extremal choices, and probabalistic methods will be emphasized with a view towards building an expertise in working in discrete applied mathematics. (3-0-3)

Prerequisite: MATH 454 or instructor's consent.

**» Sample Syllabus**

### MATH 554 Discrete Applied Mathematics II

A graduate-level course that introduces students in applied mathematics, computer science, natural sciences, and engineering, to the application of modern tools and techniques from various fields of mathematics to existential and algorithmic problems arising in discrete applied math. Probabilistic methods, entropy, linear algebra methods, Combinatorial Nullstellensatz, and Markov chain Monte Carlo, are applied to fundamental problems like Ramsey-type problems, intersecting families of sets, extremal problems on graphs and hypergraphs, optimization on discrete structures, sampling and counting discrete objects, etc. (3-0-3)

Prerequisites: MATH 332 and (MATH 454 or MATH 475) and (MATH 454 or MATH 553), or instructor's consent based on familiarity with linear algebra, probability, and discrete math.

**» Sample Syllabus**

### MATH 555 Tensor Analysis

Development of the calculus of tensors with applications to differential geometry and the formulation of the fundamental equations in various fields. (3-0-3)

Prerequisites: MATH 332 and either MATH 400 or instructor's consent.

### MATH 556 Metric Spaces

Point-set theory, compactness, completeness, connectedness, total boundedness, density, category, uniform continuity and convergence, Stone-Weierstrass theorem, fixed point theorems. (3-0-3)

Prerequisite: MATH 400.

**» Sample Syllabus**

### MATH 557 Probabilistic Methods in Combinatorics

Graduate level introduction to probabilistic methods, including linearity of expectation, the deletion method, the second moment method and the LovaszLocal Lemma. Many examples from classical results and recent research incombinatorics will be included throughout, including from Ramsey Theory, random graphs, coding theory and number theory. (3-0-3) Prerequisite: Graduate status or consent of the instructor.

**» Sample Syllabus**

### MATH 563 Mathematical Statistics

Theory of sampling distributions; principles of data reduction; interval and point estimation, sufficient statistics, order statistics, hypothesis testing, correlation and linear regression; introduction to linear models. Credit given only for MATH 425, MATH 476, MATH 525 or MATH 563. (3-0-3)

Prerequisites: MATH 475, MATH 251, and MATH 332.

**» Sample Syllabus**

### MATH 564 Applied Statistics

Linear regression and correlation models, regression parameters, prediction and confidence intervals, time series, analysis of variance and covariance. (3-0-3)

Prerequisites: MATH 332, MATH 475,and MATH 476; or instructor's consent.

**» Sample Syllabus**

### MATH 565 Monte Carlo Methods in Finance

In addition to the theoretical constructs in financial mathematics, there are also a range of computational/simulation techniques that allow for the numerical evaluation of a wide range of financial securities. This course will introduce the student to some such simulation techniques, known as Monte Carlo methods, with focus on applications in financial risk management. Monte Carlo and Quasi Monte Carlo techniques are computational sampling methods which track the behavior of the underlying securities in an option or portfolio and determine the derivative's value by taking the expected value of the discounted payoffs at maturity. Recent developments with parallel programming techniques and computer clusters have made these methods widespread in the finance industry. (3-0-3) Prerequisite: MATH 474.

**» Sample Syllabus**

### MATH 566 Multivariate Analysis

Random vectors, sample geometry and random sampling, generalized variance, multivariate normal and Wishart distributions, estimation of mean vector, confidence region, Hotelling's T^{2}, covariance, principal components, factor analysis, discrimination, clustering. (3-0-3)

Prerequisites: MATH 532, MATH 563, and MATH 564.

**» Sample Syllabus**

### MATH 567 Advanced Design of Experiments

Various type of designs for laboratory and computer experiments, including fractional factorial designs, optimal designs and space filling designs. (3-0-3)

Prerequisites: MATH 476 or MATH 474.

**» Sample Syllabus**

### MATH 568 Topics in Statistics

Categorical data analysis, contingency tables, log-linear models, nonparametric methods, sampling techniques. (3-0-3)

Prerequisite: MATH 563.

### MATH 569 Statistical Learning

The wealth of observational and experimental data available provides great opportunities for us to learn more about our world. This course teaches modern statistical methods for learning from data, such as, regression, classification, kernel methods, and support vector machines. (3-0-3)

Prerequisites: MATH 350 and MATH 474 or 475, or consent of the instructor.

**» Sample Syllabus**

### MATH 570 Data Science Seminar

Various research topics or industrial application on Data Science are presented in this seminar. (1-0-1)

**» Sample Syllabus**

### MATH 571 Data Preparation and Analysis

This course surveys industrial and scientific applications of data analytics, with case studies, including exploration of ethical issues. Students will learn how to prepare data for analysis, perform exploratory data analysis, and develop meaningful data visualizations. They will work with a variety of real world data sets and learn how to prepare data sets for analysis by cleaning and reformatting. Students will also learn to apply a variety of different data exploration techniques including summary statistics and visualization methods. (3-0-3)

**» Sample Syllabus**

### MATH 572 Data Science Practicum

In this project-oriented course, students will work in small groups to solve real-world data analysis problems and communicate their results. Innovation and clarity of presentation will be key elements of evaluation. Students will have an option to do this as an independent data analytics internship with an industry partner. (6-0-6)

**» Sample Syllabus**

### MATH 574 Bayesian Computational Statistics

Rigorous introduction to the theory of Bayesian Statistical Inference and Data Analysis, including prior and posterior distributions, Bayesian estimation and testing, Bayesian computation theories and methods, and implementation of Bayesian computation methods using popular statistical software.

**» Sample Syllabus**

### MATH 577 Computational Mathematics I

Fundamentals of matrix theory; least squares problems; computer arithmetic, conditioning and stability; direct and iterative methods for linear systems; eigenvalue problems. (3-0-3)

Prerequisite: An undergraduate numerical course such as MATH 350, or consent of the instructor.

**» Sample Syllabus**

### MATH 578 Computational Mathematics II

Polynomial interpolation; numerical solution of initial value problems for ordinary differential equations by single and multi-step methods, Runge-Kutta, Predictor-Corrector; numerical solution of boundary value problems for ordinary differential equations by shooting method, finite differences and spectal methods. (3-0-3) Credit may not be granted for both MATH 578 and MATH 478.

Prerequisite: An undergraduate numerical course such as MATH 350, or consent of the instructor.

**» Sample Syllabus**

### MATH 579 Complexity of Numerical Problems

This course is concerned with a branch of complexity theory. It studies the intrinsic complexity of numerical problems, that is, the minimum effort required for the approximate solution of a given problem up to a given error. Based on a precise theoretical foundation, lower bounds are established, i.e. bounds that hold for all algorithms. We also study the optimality of known algorithms, and describe ways to develop new algorithms if the known ones are not optimal. (3-0-3)

Prerequisite: MATH 350.

**» Sample Syllabus**

### MATH 581 Finite Elements

Various elements, error estimates, methods for solving systems of linear equations including multigrid, discontinuous Galerkin methods. Applications. (3-0-3)

Prerequisite: Undergraduate courses in numerical methods (such as Math 350) and in partial differential equations (such as Math 489), or consent of instructor.

**» Sample Syllabus**

### MATH 582 Mathematical Finance II

This course is a continuation of Math 485/548. It introduces the student tomodern continuous time mathematical finance. The major objective of the course is to present main mathematical methodologies and models underlying the area of financial engineering, and, in particular, those that provide a formal analytical basis for valuation and hedging of financial securities. (3-0-3)

Prerequisites: MATH 485/548; MATH 481/542, or consent of the instructor.

**» Sample Syllabus**

### MATH 586 Theory and Practice of Fixed Income Modelling

The course covers basics of the modern interest rate modeling and fixed income asset pricing. The main goal is to develop a practical understanding of the core methods and approaches used in practice to model interest rates and to price and hedge interest rate contingent securities. The emphasis of the course is practical rather than purely theoretical. A fundamental objective of the course is to enable the students to gain a hand-on familiarity with and understanding of the modern approaches used in practice to model interest rate markets. (3-0-3)

Prerequisites: MATH 543 or MATH 544, and MATH 485 or equivalent, or instructor's consent.

**» Sample Syllabus**

### MATH 587 Theory and practice of modeling risk and credit derivatives

This is an advanced course in the theory and practice of credit risk and credit derivatives. Students will get acquainted with structural and reduced form approaches to mathematical modeling of credit risk. Various aspects of valuation and hedging of defaultable claims will be presented. In addition, valuation and hedging of vanilla credit derivatives, such as credit default swaps, as well as vanilla credit basket derivatives, such as collateralized credit obligations, will be discussed. (3-0-3)

Prerequisite: MATH 582 or equivalent.

**» Sample Syllabus**

### MATH 589 Numerical Methods for Partial Differential Equations

The course introduces numerical methods, especially the finite difference method for solving different types of partial differential equations. The main numerical issues such as convergence and stability will be discussed. It also includes an introduction to the finite volume method, finite element method and spectral method. (3-0-3).

Prerequisites: Undergraduate courses in numerical methods (such as Math 350) and in partial differential equations (such as Math 489), or consent of the instructor.

**» Sample Syllabus**

### MATH 590 Meshfree Methods

Fundamentals of multivariate meshfree radial basis function and moving least squares methods; applications to multivariate interpolation and least squares approximation problems;applications to the numerical solution of partial differential equations; implementation in Matlab. (3-0-3)

Prerequisites: Some exposure to computational mathematics and advanced analysis, consent of the instructor.

**» Sample Syllabus**

### MATH 591 Research & Thesis M.S. Degree

Permission of Instructor

### MATH 592 - Internship in Applied Mathematics

This course is open only to students in the Master of Applied Mathematics program. It can be used in place of Math 523 - Case Studies and Project Management - subject to the approval of the Director of the Program. Progress reports and a final project report are required and submitted to the Director of the Program. (6-0-6)

### MATH 593 Seminar in Applied Mathematics

Current research topics presented in the department colloquia and seminars. (1-0-0)

### MATH 594 Professional Master's Projects

This course is only open to students in the Master of Applied Mathematics program. Participation as a member of a team on a significant real-world problem with required project report generation and a final project report. Prereq: MATH 523 or MATH 592 and approval of Director of Program. (3-0-3).

### MATH 595 Geometry for Teachers

18 semester hours of an undergraduate mathematics major completed, certification as a mathematics teacher or approval of the instructor. The course is focused on fundamental ideas and methods related to Euclidean and Non-Euclidean (e.g., spherical) geometries in two and three dimensions and their applications with emphasis on the use of technology (e.g., Geometer's Sketchpad or Cabri dynamic geometry software) and relevance to geometric concepts in the pre-college mathematics curriculum context. Various problem-solving approaches and strategies will be emphasized based on posing hypotheses, their experimental testing and investigation, the use of formal axiomatic systems to construct and analyze proofs of the corresponding geometric theorems, and visual interpretations of the results. Participants will also complete an independent study module on some aspect of Non-Euclideangeometry not addressed in the course (e.g., read and report on the book, Flatland). The course is designed as a mathematics course for graduate students in the mathematics education and certification option programs, and for practicing secondary mathematics teachers.

### MATH 596 Math for Teachers: Elementary

Certification as Mathematics teacher or approval of the instructor. An in-service workshop for pre-college teachers emphasizing the phenomenological approach to the teaching of Mathematics. (Credit: variable)

### MATH 597 Reading & Special Projects

(Credit: Variable)

### MATH 598 Math for teachers: High School

Certification as teacher or approval of instructor. An in-service workshop for pre-college teachers emphasizing the phenomenonological approach to teaching of integrated mathematics and science at the high school level. (Credit: variable)

### MATH 599 TA Training

This course provides the foundation of how to teach mathematics in the context of introductory undergraduate courses. The course is designed to encourage participation and cooperation among the graduate students, to help them prepare for a career in academia, and to help convey the many components of effective teaching. (1-0-0)

### MATH 601 Advanced Topics in Combinatorics

Course content is variable and reflects current research in combinatorics. (3-0-3)

### MATH 602 Advanced Topics in Graph Theory

Content is variable and reflects current research in graph theory. (3-0-3)

### MATH 603 Advanced Topics in Computational Mathematics

Course content is variable and reflects current research in computational mathematics. (3-0-3)

### MATH 604 Advanced Topics in Applied Analysis

Course content is variable and reflects current research in applied analysis. (3-0-3)

### MATH 605 Advanced Topics in Stochastic Analysis

course content is variable and reflects current research in stochastic analysis. (3-0-3)

### MATH 691 Research and Thesis for Ph.D.

(Variable credit)

### MFIN 501 Computational Finance I

This course describes the concepts, mathematics and programming code behind the construction of partial differential equations used in the numerical evaluation of the value of financial derivatives. Topics covered in the course include portfolio replication, risk-neutral pricing, binomial and trinomial trees, and finite difference methods. (3-0-3)

### MFIN 502 Computational Finance II

This course describes the concepts, mathematics and programming code behind the construction of stochastic differential equations (SDE's) used in the numerical evaluation of the value of financial derivatives. The course covers the implementation of SDEs using stochastic Taylor series expansions, recursive quadrature and pathe integral methods. Topics covered in the course include stochastic processes, Stratonovich stochastic calculus, Ito calculus, path integral methods and symbolic programming. (3-0-3)